↑

Properties

Label 1-3520-3520.1539-r0-0-0
Degree $1$
Conductor $3520$
Sign $-0.634 + 0.773i$
Analytic cond. $16.3468$
Root an. cond. $16.3468$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.923 − 0.382i)3-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)9-s + (0.382 − 0.923i)13-s + i·17-s + (−0.382 + 0.923i)19-s + (0.923 − 0.382i)21-s + (−0.707 − 0.707i)23-s + (−0.382 − 0.923i)27-s + (−0.923 − 0.382i)29-s + 31-s + (0.382 + 0.923i)37-s + (−0.707 + 0.707i)39-s + (0.707 + 0.707i)41-s + (0.923 − 0.382i)43-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)3-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)9-s + (0.382 − 0.923i)13-s + i·17-s + (−0.382 + 0.923i)19-s + (0.923 − 0.382i)21-s + (−0.707 − 0.707i)23-s + (−0.382 − 0.923i)27-s + (−0.923 − 0.382i)29-s + 31-s + (0.382 + 0.923i)37-s + (−0.707 + 0.707i)39-s + (0.707 + 0.707i)41-s + (0.923 − 0.382i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.634 + 0.773i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ -0.634 + 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2062381246 + 0.4360538779i\)
\(L(\frac12)\) \(\approx\) \(0.2062381246 + 0.4360538779i\)
\(L(1)\) \(\approx\) \(0.6681898205 + 0.04791483752i\)
\(L(1)\) \(\approx\) \(0.6681898205 + 0.04791483752i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (0.382 - 0.923i)T \)
17 \( 1 + iT \)
19 \( 1 + (-0.382 + 0.923i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (-0.923 - 0.382i)T \)
31 \( 1 + T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (0.923 - 0.382i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.923 + 0.382i)T \)
59 \( 1 + (-0.382 - 0.923i)T \)
61 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 + (-0.923 - 0.382i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (0.382 - 0.923i)T \)
89 \( 1 + (-0.707 + 0.707i)T \)
97 \( 1 + T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.32305949487164107556555623885, −17.74865048906734806312051008011, −17.033303245340890144369556320153, −16.38450407120404434430049262006, −15.94631579891646645646179849594, −15.31367341883377876354483725136, −14.2412780467867702409603150567, −13.619509981266453896487630416, −12.91820234249176699680516654080, −12.15194373695434076916209420114, −11.3626960302430039296126188152, −10.95864044987165999201780018791, −10.082556698185547651540354274286, −9.43976008647504245752651328324, −8.94117715191026415940743214364, −7.54871814380763598727609780272, −7.03664756012736141619756703813, −6.3289989063808047646154578409, −5.65911668716332816133065384962, −4.69282670270127596818727105113, −4.11158677071471871691984503092, −3.398383627017960902542146438710, −2.27234145959350359227346920266, −1.1142848817176491995689158415, −0.2001887692323318495915992855, 1.01419911089418127788633476072, 1.96478300245502463612758177088, 2.836142812517227990596887267434, 3.84360147880464550317802795385, 4.63150926633880073155437134419, 5.72680771924733551193134415744, 6.05236141622377054753408037059, 6.56514982489742853969816049244, 7.83117677823687676713649211642, 8.133322619589420280709469451504, 9.21696752530873957814931065486, 10.13919471393662258967503040879, 10.5072706689598291833582589430, 11.42128882378726560832829405678, 12.11604329620209364413449096904, 12.869217303607033621878036100180, 12.97338684863651531238316538479, 14.12522988794870761544601747023, 15.00782689175711772077872778114, 15.68854320892496843910239557320, 16.24181531854434428732780619123, 17.00396853160120895576053601282, 17.58153986949470192195275621546, 18.329522856481945816758863306601, 18.936374966061276690786496896564

Graph of the $Z$-function along the critical line